The aim of this paper is to investigate the theoretical approach for solvability of impulsive abstract Cauchy problem for impulsive nonlinear fractional order partial differential equations with nonlocal conditions, where the nonlinear extensible beam equation is a particular application case of this problem.

     In this paper, some conditions to guarantee the existence of bounded solution to the second order multi delayed arguments differential equation are given. The Krasnoselskii theorem used to the Lebesgue’s dominated convergence and fixed point to obtain some new sufficient conditions for existence of solutions. Some important lemmas are established that are useful to prove the main results for oscillatory property. We also submitted some sufficient conditions to ensure the oscillation criteria of bounded solutions to the same equation.

     This research aims to present some  results for  conceptions of  quasi -hyponormal operator defined on Hilbert space . Signified by the -operator, together with some significant characteristics of this operator and various theorems pertaining to this operator are discussed, as well as, we discussed the null space and range of these  kinds of operators.

In this paper, some necessary and sufficient conditions are obtained to ensure the oscillatory of all solutions of the first order impulsive neutral differential equations. Also, some results in the references have been improved and generalized. New lemmas are established to demonstrate the oscillation property. Special impulsive conditions associated with neutral differential equation are submitted. Some examples are given to illustrate the obtained results.

     . Suppose that  is the Cayley graph whose vertices are all elements of  and two vertices  and  are adjacent if and only if . In this paper,we introduce the generalized Cayley graph denoted by  which is a graph with a vertex set consisting of all column matrices  in which all components are in  and two vertices  and  are adjacent if and only if , where  is a column matrix that each entry is the inverse of the similar entry of  and  is  matrix with all entries in  ,  is the transpose of  and  and m . We aim to provide some basic properties of the new graph and determine the structure of  when  is a complete graph  for every , and n, m  .

     Fractional calculus has paid much attention in recent years, because it plays an essential role in many fields of science and  engineering, where the study of stability theory of fractional differential equations emerges to be very important. In this paper, the stability of fractional order ordinary differential equations will be studied and introduced the backstepping method. The Lyapunov function  is easily found by this method. This method also gives a guarantee of stable solutions for the fractional order differential equations. Furthermore it gives asymptotically stable.

The aim of this paper is to introduce a certain family of new classes of multivalent functions associated with subordination. The various results obtained here for each of these classes include coefficient estimates radius of convexity, distortion and growth theorem.  

In this paper, making use of the q-R uscheweyh differential  operator , and  the  notion of t h e J anowski f unction, we study some subclasses of  holomorphic   f- unction s . Moreover , we obtain so me geometric characterization like co efficient es timat es , rad ii of starlikeness ,distortion theorem , close- t o- convexity , con vexity, ext reme point s, neighborhoods, and the i nte gral mean inequalities of func tions affiliation to these c lasses

A new Differential Evolution (ARDE) algorithm is introduced that automatically adapt a repository of DE strategies and parameters adaptation schemes of the mutation factor and the crossover rate to avoid the problems of stagnation and make DE responds to a wide range of function characteristics at different stages of the evolution. ARDE algorithm makes use of JADE strategy and the MDE_pBX parameters adaptive schemes as frameworks. Then a new adaptive procedure called adaptive repository (AR) has been developed to select the appropriate combinations of the JADE strategies and the parameter control schemes of the MDE_pBX to generate the next population based on their fitness values. Experimental results have been presented to confirm the reli

     The idea of the paper is to consolidate Mahgoub transform and variational iteration method (MTVIM) to solve fractional delay differential equations (FDDEs). The fractional derivative was in Caputo sense. The convergences of approximate solutions to exact solution were quick. The MTVIM is characterized by ease of application in various problems and is capable of simplifying the size of computational operations.  Several non-linear (FDDEs) were analytically solved as illustrative examples and the results were compared numerically. The results for accentuating the efficiency, performance, and activity of suggested method were shown by comparisons with Adomian Decomposition Method (ADM), Laplace Adomian Decompos